Definition 2.3.1 (The semigroup of projections of A):

Put where is a C-star-Algebra and . We view the sets as being pair-wise disjoint.
Define the relation on as follows.
Suppose that is a projection in and is a projection in Then if and only if there exists an element with and . Here is the set of matrices with entries in . The adjoint is the entry-wise adjoint, and products are well-defined for these objects.